Dirac series for complex classical Lie groups: A multiplicity-one theorem

TL;DR

This paper calculates the Dirac cohomology of irreducible unitary modules for complex classical groups, showing that those with nonzero cohomology are induced from unipotent representations and have a unique contributing $K$-type.

Contribution

It confirms conjectures about the structure of Dirac cohomology in unitary representations of complex classical groups, specifically regarding multiplicity and induction from unipotent representations.

Findings

Unitary representations with nonzero Dirac cohomology are induced from unipotent representations.

Such representations have a unique, multiplicity-free $K$-type contributing to the Dirac cohomology.

The results confirm conjectures from 2011 about the structure of Dirac cohomology.

Abstract

This paper computes the Dirac cohomology $H_D(\pi)$ of irreducible unitary Harish-Chandra modules $\pi$ of complex classical groups viewed as real reductive groups. More precisely, unitary representations with nonzero Dirac cohomology are shown to be unitarily induced from unipotent representations. When nonzero, there is a unique, multiplicity free $K-$type in $\pi$ contributing to $H_D(\pi)$. This confirms conjectures formulated by the first named author and Pandzic in 2011.